Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70 769 Using functions of a complex variable and integrals of Cauchy type in monographs G.V. Kolosov [4] and I. I. Muskhelishvili [5] developed an effective method for solving plane boundary problems of elasticity, which later became classical. Although this method is not directly applicable to the spatial problems of elasticity, the apparatus of functions of a complex variable is also used in solving problems of this class. So, G. N. Polozhii [6] proposed a method for solving axisymmetric spatial problems in elasticity, it involves using two P -analytic functions and is an analog of Kolosov - Muskhelishvili's complex potential method. Another way to investigate three-dimensional problems is the method of integral equations, with its help, the existence and unique theorems for the solution of boundary value problems can be proved. This method often serves as a basis for developing algorithms for the numerical solution of elasticity problems. The monograph [7] is devoted to the method of integral equations. The method of integral transforms is widely used in spatial problems of elasticity. With the help of the corresponding integral transform, a transition is made to a simpler problem in the domain of transforms. An extensive bibliography of papers on the use of this method in problems of elasticity theory is given in the monograph by Ya. S. Uflyand [8]. Thus, the above methods, using different representations’ solutions through auxiliary harmonic and biharmonic functions, which, on one hand, facilitated solving in terms of these functions, but on the other hand it causes difficulties during the calculation of the original functions These difficulties in itself present a complex mathematical problem. One of the basic methods to construct an exact solution for spatial problems of elasticity is the Fourier method, which is based on the use of curvilinear orthogonal coordinate systems that allow the separation of variables in the three- dimensional Laplace equation for static problems. One notes that solutions obtained by the Fourier method are the initial steps to construct solutions for problems of finite bodies bounded by surfaces described by canonical coordinate systems. Using the method of variable separation demands different representations of the equilibrium equations solutions through the stress functions. With the help of such representations, the original problem is reduced to solving differential equations of a simpler structure. Each stress function in these equations is not related to the others, but it is presented in the boundary conditions together with the others. The solution in the form of Papkovich - Neuber is used most often, because it allows the application of classical solutions of the potential theory by solving boundary value problems, represented in the form of series and integrals containing special functions. Another way to investigate spatial problems is the method of eigenvector functions, which is a vector analog of the Fourier method, this was proposed in [9]. It involves the construction of vector structure eigen functions on the boundary surface of the bodies. With the help of this method, solutions were obtained for complex three-dimensional problems of elasticity. In paper [10] a method of two-dimensional singular equations for three-dimensional problems of stationary thermal conductivity and thermoelasticity for bodies with cracks was developed, and several problems for a semi-infinite body with cracks were solved. A new approach proposed by G. Ya. Popov in [11] is used in this paper. The method is based on reducing Lame equations to one independently solvable and two jointly solved equations. Moreover, the boundary conditions are also separated, which greatly simplifies the calculation technique in comparison with traditional methods. By the method of integral transforms applied directly to the transformed equations of equilibrium and the boundary conditions of the initial stated problem, one reduces it to a one-dimensional vector boundary value problem, which is solved exactly. The problem of elasticity for a quarter space was solved with this method in [12]. The elastic layer, as an important and frequently used model object, has been studied by many authors. Thus, in a static formulation, the axisymmetric contact problems for an elastic layer and strip are considered in [13-15], various mixed boundary-value thermoelasticity problems for a half-layer and a half-space in [16-23], layers with thermal loadings were considered in [24] with no static loading, where the conformal mapping method was used, dynamic problems were investigated in [25-27]. The papers [28, 29] are devoted to solving elasticity problems for bodies with their proper weight. In paper [30] the authors investigated the propagation of thermoelastic waves through the layers and analyzed the importance of thermally nonlinear generalized thermoelastic analysis. In [31] the size-dependent quality factor of thermoelastic damping in a microbeam resonator, based on modified strain gradient elasticity was analyzed. The generalized thermoelasticity theory of the Lord–Shulman model was used to derive the equation of coupled thermoelasticity. The paper [32] is devoted to study functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, where the effects of external load at the inner and outer radii on radial displacement as well as stress components were considered. In [33] the influence of measurement errors on the accuracy of the estimated heat flux and mechanical load on laminated functionally graded plate was investigated. Thermo-elasticity problems of functionally graded materials were evaluated in [34]. In [35] an exact solution for thermoelastic deformations of functionally graded thick rectangular plates was derived. The contact problem of two semi-infinite Kirchhoff plates on the